c Oksana Shatalov, Spring 2016 1 4&5 Binary Operations and Relations. The Integers. (part I) 4.1: Binary Operations DEFINITION 1. A binary operation ∗ on a nonempty set A is a function from A × A to A. Addition, subtraction, multiplication are binary operations on Z. Addition is a binary operation on Q because Division is NOT a binary operation on Z because Division is a binary operation on • To prove that ∗ is a binary operation on a set A • To show that ∗ is not a binary operation on a set A c Oksana Shatalov, Spring 2016 2 Classification of binary operations by their properties Associative and Commutative Laws DEFINITION 2. A binary operation ∗ on A is associative if 8a; b; c 2 A; (a ∗ b) ∗ c = a ∗ (b ∗ c): A binary operation ∗ on A is commutative if 8a; b 2 A; a ∗ b = b ∗ a: EXAMPLE 3. Using symbols complete the following (a) A binary operation ∗ on A is not associative if (b) A binary operation ∗ on A is not commutative if Identities DEFINITION 4. If ∗ is a binary operation on A, an element e 2 A is an identity element of A w.r.t ∗ if 8a 2 A; a ∗ e = e ∗ a = a: EXAMPLE 5. (a) 1 is an identity element for Z, Q and R w.r.t. multiplication. (b) 0 is an identity element for Z, Q and R w.r.t. addition. Inverses DEFINITION 6. Let ∗ be a binary operation on A with identity e, and let a 2 A. We say that a is invertible w.r.t. ∗ if there exists b 2 A such that a ∗ b = b ∗ a = e: If b exists, we say that b is an inverse of a w.r.t. ∗ and write b = a−1: Note, inverses may or may not exist. c Oksana Shatalov, Spring 2016 3 EXAMPLE 7. Every x 2 Z has inverse w.r.t. addition because 8x 2 Z; x + (−x) = (−x) + x = 0: However, very few elements in Z have multiplicative inverses. Namely, EXAMPLE 8. Let ∗ be an operation on Z defined by 8a; b 2 Z; a ∗ b = a + 3b − 1: (a) Prove that the operation is binary. (b) Determine whether the operation is associative and/or commutative. Prove your answers. (c) Determine whether the operation has identities. (d) Discuss inverses. c Oksana Shatalov, Spring 2016 4 EXAMPLE 9. Let ∗ be an operation on the power set P (A) defined by 8X; Y 2 P (A);X ∗ Y = X \ Y: (a) Prove that the operation is binary. (b) Determine whether the operation is associative and/or commutative. Prove your answers. (c) Determine whether the operation has identities. (d) Discuss inverses. c Oksana Shatalov, Spring 2016 5 EXAMPLE 10. Let ∗ be an operation on F (A) defined by 8f; g 2 F (A); f ∗ g = f ◦ g: (a) Prove that the operation is binary. (b) Determine whether the operation is associative and/or commutative. (c) Determine whether the operation has identities. (d) Discuss inverses. PROPOSITION 11. Let ∗ be a binary operation on a nonempty set A. If e is an identity element on A then e is unique. Proof. c Oksana Shatalov, Spring 2016 6 PROPOSITION 12. Let ∗ be an associative binary operation on a nonempty set A with the identity e, and if a 2 A has an inverse element w.r.t. ∗, then this inverse element is unique. Proof. Closure DEFINITION 13. Let ∗ be a binary operation on a nonempty set A, and suppose that S ⊆ A. If ∗ is also a binary operation on S then we say that S is closed in A under ∗. EXAMPLE 14. Let ∗ be a binary operation on A and let S ⊆ A. Using symbols complete the following (a) S is closed in A under ∗ if and only if (a) S is not closed in A under ∗ if EXAMPLE 15. Determine whether the following subsets of Z are closed in Z under addition and mul- tiplication. (a) Z+ (b) E (c) O c Oksana Shatalov, Spring 2016 7 5.1: The Integers: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition is associative: A2. Addition is commutative: A3. Z has an identity element with respect to addition namely, the integer 0. A4. Every integer x in Z has an inverse w.r.t. addition, namely, its negative −x : A5. Multiplication is associative: A6. Multiplication is commutative: A7. Z has an identity element with respect to multiplication namely, the integer 1. (and 1 6= 0:) A8. Distributive Law: REMARK 16. We do not prove A1-A8. We take them as axioms: statements we assume to be true about the integers. We use xy instead x · y and x − y instead x + (−y): PROPOSITION 17. Let a; b; c 2 Z: P1. If a + b = a + c then b = c: (cancellation law for addition) P2. a · 0 = 0 · a = 0: P3. (−a)b = a(−b) = −(ab) P4. −(−a) = a P5. (−a)(−b) = ab P6. a(b − c) = ab − ac P7. (−1)a = −a P8. (−1)(−1) = 1: c Oksana Shatalov, Spring 2016 8 Proof Z contains a subset Z+, called the positive integers, that has the following properties: A9. Closure property: Z+ is closed w.r.t. addition and multiplication: A10. Trichotomy Law: for all x 2 Z exactly one is true: PROPOSITION 18. If x 2 Z, x 6= 0, then x2 2 Z+: Proof. COROLLARY 19. Z+ = f1; 2; 3; : : : ; n; n + 1;:::g Proof. c Oksana Shatalov, Spring 2016 9 Inequalities (the order relation less than) DEFINITION 20. For x; y 2 Z, x < y if and only y − x 2 Z+: REMARK 21. If x < y, we can also write y > x. We can also write x ≤ y if x < y or x = y. Similarly, y ≥ x if y > x or y = x. Note that Z+ = fn 2 Zjn > 0g : EXAMPLE 22. Let x; y 2 Z. Using symbols complete the following • x < y , • x > y , • x < 0 , • x > 0 , PROPOSITION 23. Let a; b 2 Z. Q1. Exactly one of the following holds: a < b, b < a, or a = b: Q2. If a > 0 then −a < 0; if a < 0 then −a > 0: Q3. If a > 0 and b > 0 then a + b > 0 and ab > 0: Q4. If a > 0 and b < 0 then ab < 0. Q5. If a < 0 and b < 0 then ab > 0. Proof. c Oksana Shatalov, Spring 2016 10 PROPOSITION 24. Let a; b; c 2 Z. Q6. If a < b and b < c then a < c. Q7. If a < b and a + c < b + c. Q8. If a < b and c > 0 then ac < bc. Q9. If a < b and c < 0 then ac > bc. A11. The Well Ordering Principle Every nonempty subset on Z+ has a smallest element; that is, if S is a nonempty subset of Z+, then there exists a 2 S such that a ≤ x for all x 2 S: PROPOSITION 25. There is no integer x such that 0 < x < 1: Proof. COROLLARY 26. 1 is the smallest element of Z+: COROLLARY 27. The only integers having multiplicative inverses in Z are ±1: c Oksana Shatalov, Spring 2016 11 5.2: Induction1 THEOREM 28. (First Principle of Mathematical Induction) Let P (n) be a statement about the positive integer n. Suppose that P (1) is true. Whenever k is a positive integer for which P (k) is true, then P (k + 1) is true. Then P (n) is true for every positive integer n. Proof. Paradox: All horses are of the same color. Question: What's wrong in the following \proof" of G. P´olya? P (n) : Let n 2 Z+. Within any set of n horses, there is only one color. Basic Step. If there is only one horse, there is only one color. Induction Hypothesis. Assume that within any set of k horses, there is only one color. Inductive step. Prove that within any set of k + 1 horses, there is only one color. Indeed, look at any set of k + 1 horses. Number them: 1; 2; 3; :::; k; k + 1: Consider the subsets f1; 2; 3; :::; kg and f2; 3; 4; :::; k + 1g. Each is a set of only k horses, therefore within each there is only one color. But the two sets overlap, so there must be only one color among all k + 1 horses. 1see also Chapter 1(Part III).